The ropelengths of knots are almost linear in terms of their crossing numbers

摘要

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a > 0 such that L(K) <= a Cr(K) ln(5)(Cr(K)) for any K, i.e. the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number. This result is a significant improvement over the best known upper bound established previously, which is of the form L(K) <= O(Cr(K) 3/2). The proof is based on a divide-and-conquer approach on 4-regular plane graphs: a 4-regular plane graph of n is first repeatedly subdivided into many small subgraphs and then reconstructed from these small subgraphs on the cubic lattice with its topology preserved with a total length of the order O(n ln(5)(n)). The result then follows since a knot can be recovered from a graph that is topologically equivalent to a regular projection of it (which is a 4-regular plane graph).